An Introduction to Riemannian Geometry by C. E. Weatherburn

By C. E. Weatherburn

The aim of this e-book is to bridge the distance among differential geometry of Euclidean area of 3 dimensions and the extra complicated paintings on differential geometry of generalised area. the topic is handled through the Tensor Calculus, that is linked to the names of Ricci and Levi-Civita; and the ebook offers an creation either to this calculus and to Riemannian geometry. The geometry of subspaces has been significantly simplified via use of the generalized covariant differentiation brought by means of Mayer in 1930, and effectively utilized by way of different mathematicians.

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By Bp we denote the restriction Bp = B|⊗rl=1 Tp M of B to the r-fold tensor product of Tp M given by Bp : ((X1 )p , . . , (Xr )p ) → B(X1 , . . , Xr )(p). 2. Let M be a smooth manifold. A Riemannian metric on M is a tensor field g : C2∞ (T M) → C0∞ (T M) such that for each p ∈ M the restriction gp = g|Tp M ⊗Tp M : Tp M ⊗ Tp M → R with gp : (Xp , Yp ) → g(X, Y )(p) is an inner product on the tangent space Tp M. The pair (M, g) is called a Riemannian manifold. The study of Riemannian manifolds is called Riemannian Geometry.

For this we write X part (X ∞ ˜ Y˜ ∈ C (T M) be vector fields on M and X, Y ∈ C ∞ (T N) Let X, be their extensions to N. If p ∈ M then (∇XY )p only depends on the ˜ p and the value of Y along some curve γ : (− , ) → N value Xp = X ˜ p . 3. Since such that γ(0) = p and γ(0) ˙ = Xp = X Xp ∈ Tp M we may choose the curve γ such that the image γ((− , )) is contained in M. Then Y˜γ(t) = Yγ(t) for t ∈ (− , ). This means ˜ p and the value of Y˜ along γ, hence that (∇XY )p only depends on X ˜ and Y˜ are extended.

As for the orthogonal group O(m) an inner product on the tangent space at the neutral element of any Lie group can be transported via the left translations to obtain a left invariant Riemannian metric on the group. 14. Let G be a Lie group and , e be an inner product on the tangent space Te G at the neutral element e. Then for each x ∈ G the bilinear map gx (, ) : Tx G × Tx G → R with gx (Xx , Yx ) = dLx−1 (Xx ), dLx−1 (Yx ) e is an inner product on the tangent space Tx G. The smooth tensor field g : C2∞ (T G) → C0∞ (G) given by g : (X, Y ) → (g(X, Y ) : x → gx (Xx , Yx )) is a left invariant Riemannian metric on G.

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